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I'm having problems with fundementals of partial derivatives. I've looked around for something like "partial derivative FAQ", but couldn't find.

Can any of the following be true in general?

[tex]\frac{\partial f}{\partial x} = \frac{\partial f}{\partial g}\frac{\partial g}{\partial x}[/tex]

[tex]\frac{d f}{d x} = \frac{\partial f}{\partial g}\frac{\partial g}{\partial x}[/tex]

I'd rather expect

[tex]\frac{d f}{d x} = \frac{d f}{d g}\frac{d g}{d x}[/tex]

but the partial version seems to be working equally. Can there be cases where the partial version fails?

Also, it's weird that although

[tex]\frac{d f}{d g} = \frac{1}{\frac{d g}{d f}}[/tex]

holds for all,

[tex]\frac{\partial f}{\partial g} = \frac{1}{\frac{\partial g}{\partial f}}[/tex]

doesn't. Why not?

I'm desperately looking for aprooffor any of them...

p.s.: You may ask which one is independent variable and on what f and g depends. I leave it to the guy who's willing to prove them, because I'm not sure whether if makes any difference in a general proof.

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# Partial derivate philosophy

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